Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

From Configuration-Space Clearance to Feature-Space Margin: Sample Complexity in Learning-Based Collision Detection

Sapir Tubul, Aviv Tamar, Kiril Solovey, Oren Salzman

TL;DR

This work addresses the lack of formal guarantees for learning-based collision detection (LCD) in robot motion planning by deriving a sample-complexity framework for an SVM-based LCD. It introduces a δ-centered, grid-based feature mapping φ_σ that yields a feature-space margin, and proves a bound on the required sample size m_{X^δ}(ε, ξ) to ensure a specified misclassification rate on the δ-interior of the configuration space. The authors connect C-space clearance to a feature-space margin, enabling a Hard-SVM-based LCD with statistical guarantees that hold with probability at least 1−ξ. They also propose an adaptive learning algorithm to select δ and m to meet a target ε, ξ, and provide empirical results illustrating the trade-offs and practical considerations. While the framework offers formal guarantees, it warns about exponential dependence on the dimension and inverse clearance, suggesting future work to improve practicality and extend guarantees to broader regions of the C-space.

Abstract

Motion planning is a central challenge in robotics, with learning-based approaches gaining significant attention in recent years. Our work focuses on a specific aspect of these approaches: using machine-learning techniques, particularly Support Vector Machines (SVM), to evaluate whether robot configurations are collision free, an operation termed ``collision detection''. Despite the growing popularity of these methods, there is a lack of theory supporting their efficiency and prediction accuracy. This is in stark contrast to the rich theoretical results of machine-learning methods in general and of SVMs in particular. Our work bridges this gap by analyzing the sample complexity of an SVM classifier for learning-based collision detection in motion planning. We bound the number of samples needed to achieve a specified accuracy at a given confidence level. This result is stated in terms relevant to robot motion-planning such as the system's clearance. Building on these theoretical results, we propose a collision-detection algorithm that can also provide statistical guarantees on the algorithm's error in classifying robot configurations as collision-free or not.

From Configuration-Space Clearance to Feature-Space Margin: Sample Complexity in Learning-Based Collision Detection

TL;DR

This work addresses the lack of formal guarantees for learning-based collision detection (LCD) in robot motion planning by deriving a sample-complexity framework for an SVM-based LCD. It introduces a δ-centered, grid-based feature mapping φ_σ that yields a feature-space margin, and proves a bound on the required sample size m_{X^δ}(ε, ξ) to ensure a specified misclassification rate on the δ-interior of the configuration space. The authors connect C-space clearance to a feature-space margin, enabling a Hard-SVM-based LCD with statistical guarantees that hold with probability at least 1−ξ. They also propose an adaptive learning algorithm to select δ and m to meet a target ε, ξ, and provide empirical results illustrating the trade-offs and practical considerations. While the framework offers formal guarantees, it warns about exponential dependence on the dimension and inverse clearance, suggesting future work to improve practicality and extend guarantees to broader regions of the C-space.

Abstract

Motion planning is a central challenge in robotics, with learning-based approaches gaining significant attention in recent years. Our work focuses on a specific aspect of these approaches: using machine-learning techniques, particularly Support Vector Machines (SVM), to evaluate whether robot configurations are collision free, an operation termed ``collision detection''. Despite the growing popularity of these methods, there is a lack of theory supporting their efficiency and prediction accuracy. This is in stark contrast to the rich theoretical results of machine-learning methods in general and of SVMs in particular. Our work bridges this gap by analyzing the sample complexity of an SVM classifier for learning-based collision detection in motion planning. We bound the number of samples needed to achieve a specified accuracy at a given confidence level. This result is stated in terms relevant to robot motion-planning such as the system's clearance. Building on these theoretical results, we propose a collision-detection algorithm that can also provide statistical guarantees on the algorithm's error in classifying robot configurations as collision-free or not.

Paper Structure

This paper contains 16 sections, 4 theorems, 10 equations, 3 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Learning-Based Collision Detection in Motion Planning
  4. Sample Complexity
  5. Algorithmic Background
  6. Classification
  7. Support Vector Machines
  8. Theoretical Framework
  9. Definitions
  10. Theorems
  11. Discussion
  12. LCD with statistical guarantees
  13. LCD with Guarantees---Foundation
  14. LCD with Guarantees
  15. Empirical Results
  16. ...and 1 more sections

Key Result

Theorem 1

For any $\delta {\in (0, \sqrt{d})}$, set $n: = {{\sqrt{d}}/{\delta}}$ and $\sigma^2 := {{2\delta^2}/{\ln(9n^d)}}$. Given the feature mapping $\phi_\sigma$, then the set of feature-space points $\{\phi_\sigma(x)~\vert~x\in \mathcal{X}^{\delta}_{\text{free}} \}$ and $\{\phi_\sigma(x)~\vert~x\in \math

Figures (3)

  • Figure 1: (a) Workspace with a two-link robot in three different configurations, each with its own color and label: orange (1), green (2) and purple (3). (b) C-space representation depicting $\mathcal{X}_{\text{free}}$ () and $\mathcal{X}_{\text{forb}}$ (), as well as configurations whose color corresponds to the robot's color in that configuration. Circles around each configuration correspond to its clearances. (c) $\delta\text{-interior}$$\mathcal{X}^{\delta}$ ( and ) and $\delta$-boundary $\mathcal{X}^{\overline{\delta}}$ (). Configurations (1) and (2) lie in $\mathcal{X}^{\delta}$, while (3) is in $\mathcal{X}^{\overline{\delta}}$.
  • Figure 2: Illustration of the learning process for the C-space visualized in Fig. \ref{['fig:workspace_delta']} depicting $\mathcal{X}_{\text{free}}$ () and $\mathcal{X}_{\text{forb}}$ (): (a) The $\delta$-boundary for $\delta = \delta_{\max}$ (), where Alg \ref{['algo:Learning Phase']} fails because $\varepsilon_{\mathcal{X}^{\delta}} \leq 0$ (C1 holds). (b) The $\delta$-boundary for $\delta <~\delta^*$ () and $m > m^*$ samples, that all lie in the $\delta$-interior for learning the CD, where (C1) and (C2) do not hold. (c) The space learned by the LCD with the boundary between $\mathcal{X}_{\text{forb}}$ and $\mathcal{X}_{\text{forb}}$ dashed and highlighting $\hat{\mathcal{X}_{\text{forb}}}$ (), the part estimating $\mathcal{X}_{\text{forb}}$.
  • Figure 3: Trends between key parameters in our framework. (a) The size $\vert \mathcal{X}^{\delta} \vert$ of the $\delta$-interior as a function of $\delta$. (b) Classification error $\varepsilon_{\mathcal{X}^{\delta}}$ in $\delta$-interior as a function of $\delta$ for different $\varepsilon$ values. (c) Sample complexity as a function of $\delta$ for different $\varepsilon$ values.

Theorems & Definitions (12)

  • Definition 1: Clearance
  • Definition 2: $\delta$-interior $\&$ $\delta$-boundary
  • Definition 3
  • Definition 4: margin
  • Theorem 1: Feature-Space Margin
  • proof : sketch
  • Theorem 2: Sample Complexity for learning $\mathcal{X}^{\delta}$
  • proof : sketch
  • Lemma 1: LCD existence
  • proof
  • ...and 2 more